Properties

Label 2280.2.a.p.1.2
Level $2280$
Weight $2$
Character 2280.1
Self dual yes
Analytic conductor $18.206$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.2058916609\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 2280.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +5.23607 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +5.23607 q^{7} +1.00000 q^{9} -1.23607 q^{11} +0.763932 q^{13} -1.00000 q^{15} +4.47214 q^{17} +1.00000 q^{19} +5.23607 q^{21} +2.47214 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.763932 q^{29} -8.94427 q^{31} -1.23607 q^{33} -5.23607 q^{35} +3.23607 q^{37} +0.763932 q^{39} -9.70820 q^{41} +5.23607 q^{43} -1.00000 q^{45} +2.47214 q^{47} +20.4164 q^{49} +4.47214 q^{51} -0.472136 q^{53} +1.23607 q^{55} +1.00000 q^{57} -10.4721 q^{59} +4.47214 q^{61} +5.23607 q^{63} -0.763932 q^{65} +1.52786 q^{67} +2.47214 q^{69} +12.4721 q^{73} +1.00000 q^{75} -6.47214 q^{77} +1.00000 q^{81} +4.00000 q^{83} -4.47214 q^{85} +0.763932 q^{87} -1.70820 q^{89} +4.00000 q^{91} -8.94427 q^{93} -1.00000 q^{95} -4.76393 q^{97} -1.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 6 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{19} + 6 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{33} - 6 q^{35} + 2 q^{37} + 6 q^{39} - 6 q^{41} + 6 q^{43} - 2 q^{45} - 4 q^{47} + 14 q^{49} + 8 q^{53} - 2 q^{55} + 2 q^{57} - 12 q^{59} + 6 q^{63} - 6 q^{65} + 12 q^{67} - 4 q^{69} + 16 q^{73} + 2 q^{75} - 4 q^{77} + 2 q^{81} + 8 q^{83} + 6 q^{87} + 10 q^{89} + 8 q^{91} - 2 q^{95} - 14 q^{97} + 2 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 5.23607 1.97905 0.989524 0.144370i \(-0.0461154\pi\)
0.989524 + 0.144370i \(0.0461154\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.23607 −0.372689 −0.186344 0.982485i \(-0.559664\pi\)
−0.186344 + 0.982485i \(0.559664\pi\)
\(12\) 0 0
\(13\) 0.763932 0.211877 0.105938 0.994373i \(-0.466215\pi\)
0.105938 + 0.994373i \(0.466215\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 5.23607 1.14260
\(22\) 0 0
\(23\) 2.47214 0.515476 0.257738 0.966215i \(-0.417023\pi\)
0.257738 + 0.966215i \(0.417023\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.763932 0.141859 0.0709293 0.997481i \(-0.477404\pi\)
0.0709293 + 0.997481i \(0.477404\pi\)
\(30\) 0 0
\(31\) −8.94427 −1.60644 −0.803219 0.595683i \(-0.796881\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 0 0
\(33\) −1.23607 −0.215172
\(34\) 0 0
\(35\) −5.23607 −0.885057
\(36\) 0 0
\(37\) 3.23607 0.532006 0.266003 0.963972i \(-0.414297\pi\)
0.266003 + 0.963972i \(0.414297\pi\)
\(38\) 0 0
\(39\) 0.763932 0.122327
\(40\) 0 0
\(41\) −9.70820 −1.51617 −0.758083 0.652158i \(-0.773864\pi\)
−0.758083 + 0.652158i \(0.773864\pi\)
\(42\) 0 0
\(43\) 5.23607 0.798493 0.399246 0.916844i \(-0.369272\pi\)
0.399246 + 0.916844i \(0.369272\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 2.47214 0.360598 0.180299 0.983612i \(-0.442293\pi\)
0.180299 + 0.983612i \(0.442293\pi\)
\(48\) 0 0
\(49\) 20.4164 2.91663
\(50\) 0 0
\(51\) 4.47214 0.626224
\(52\) 0 0
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) 1.23607 0.166671
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −10.4721 −1.36336 −0.681678 0.731652i \(-0.738749\pi\)
−0.681678 + 0.731652i \(0.738749\pi\)
\(60\) 0 0
\(61\) 4.47214 0.572598 0.286299 0.958140i \(-0.407575\pi\)
0.286299 + 0.958140i \(0.407575\pi\)
\(62\) 0 0
\(63\) 5.23607 0.659683
\(64\) 0 0
\(65\) −0.763932 −0.0947541
\(66\) 0 0
\(67\) 1.52786 0.186658 0.0933292 0.995635i \(-0.470249\pi\)
0.0933292 + 0.995635i \(0.470249\pi\)
\(68\) 0 0
\(69\) 2.47214 0.297610
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 12.4721 1.45975 0.729877 0.683579i \(-0.239578\pi\)
0.729877 + 0.683579i \(0.239578\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −6.47214 −0.737568
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −4.47214 −0.485071
\(86\) 0 0
\(87\) 0.763932 0.0819021
\(88\) 0 0
\(89\) −1.70820 −0.181069 −0.0905346 0.995893i \(-0.528858\pi\)
−0.0905346 + 0.995893i \(0.528858\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) −8.94427 −0.927478
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −4.76393 −0.483704 −0.241852 0.970313i \(-0.577755\pi\)
−0.241852 + 0.970313i \(0.577755\pi\)
\(98\) 0 0
\(99\) −1.23607 −0.124230
\(100\) 0 0
\(101\) −13.4164 −1.33498 −0.667491 0.744618i \(-0.732632\pi\)
−0.667491 + 0.744618i \(0.732632\pi\)
\(102\) 0 0
\(103\) 9.52786 0.938808 0.469404 0.882983i \(-0.344469\pi\)
0.469404 + 0.882983i \(0.344469\pi\)
\(104\) 0 0
\(105\) −5.23607 −0.510988
\(106\) 0 0
\(107\) −8.94427 −0.864675 −0.432338 0.901712i \(-0.642311\pi\)
−0.432338 + 0.901712i \(0.642311\pi\)
\(108\) 0 0
\(109\) −4.47214 −0.428353 −0.214176 0.976795i \(-0.568707\pi\)
−0.214176 + 0.976795i \(0.568707\pi\)
\(110\) 0 0
\(111\) 3.23607 0.307154
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −2.47214 −0.230528
\(116\) 0 0
\(117\) 0.763932 0.0706255
\(118\) 0 0
\(119\) 23.4164 2.14658
\(120\) 0 0
\(121\) −9.47214 −0.861103
\(122\) 0 0
\(123\) −9.70820 −0.875359
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 5.23607 0.461010
\(130\) 0 0
\(131\) 14.1803 1.23894 0.619471 0.785020i \(-0.287347\pi\)
0.619471 + 0.785020i \(0.287347\pi\)
\(132\) 0 0
\(133\) 5.23607 0.454025
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 15.8885 1.35745 0.678725 0.734393i \(-0.262533\pi\)
0.678725 + 0.734393i \(0.262533\pi\)
\(138\) 0 0
\(139\) −14.4721 −1.22751 −0.613755 0.789496i \(-0.710342\pi\)
−0.613755 + 0.789496i \(0.710342\pi\)
\(140\) 0 0
\(141\) 2.47214 0.208191
\(142\) 0 0
\(143\) −0.944272 −0.0789640
\(144\) 0 0
\(145\) −0.763932 −0.0634411
\(146\) 0 0
\(147\) 20.4164 1.68392
\(148\) 0 0
\(149\) 17.4164 1.42681 0.713404 0.700753i \(-0.247153\pi\)
0.713404 + 0.700753i \(0.247153\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 4.47214 0.361551
\(154\) 0 0
\(155\) 8.94427 0.718421
\(156\) 0 0
\(157\) −21.4164 −1.70922 −0.854608 0.519274i \(-0.826202\pi\)
−0.854608 + 0.519274i \(0.826202\pi\)
\(158\) 0 0
\(159\) −0.472136 −0.0374428
\(160\) 0 0
\(161\) 12.9443 1.02015
\(162\) 0 0
\(163\) 20.6525 1.61763 0.808813 0.588065i \(-0.200110\pi\)
0.808813 + 0.588065i \(0.200110\pi\)
\(164\) 0 0
\(165\) 1.23607 0.0962278
\(166\) 0 0
\(167\) 13.8885 1.07473 0.537364 0.843350i \(-0.319420\pi\)
0.537364 + 0.843350i \(0.319420\pi\)
\(168\) 0 0
\(169\) −12.4164 −0.955108
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 0 0
\(175\) 5.23607 0.395810
\(176\) 0 0
\(177\) −10.4721 −0.787134
\(178\) 0 0
\(179\) −15.4164 −1.15228 −0.576138 0.817352i \(-0.695441\pi\)
−0.576138 + 0.817352i \(0.695441\pi\)
\(180\) 0 0
\(181\) 8.47214 0.629729 0.314864 0.949137i \(-0.398041\pi\)
0.314864 + 0.949137i \(0.398041\pi\)
\(182\) 0 0
\(183\) 4.47214 0.330590
\(184\) 0 0
\(185\) −3.23607 −0.237920
\(186\) 0 0
\(187\) −5.52786 −0.404237
\(188\) 0 0
\(189\) 5.23607 0.380868
\(190\) 0 0
\(191\) −6.18034 −0.447194 −0.223597 0.974682i \(-0.571780\pi\)
−0.223597 + 0.974682i \(0.571780\pi\)
\(192\) 0 0
\(193\) 5.70820 0.410886 0.205443 0.978669i \(-0.434137\pi\)
0.205443 + 0.978669i \(0.434137\pi\)
\(194\) 0 0
\(195\) −0.763932 −0.0547063
\(196\) 0 0
\(197\) −12.4721 −0.888603 −0.444301 0.895877i \(-0.646548\pi\)
−0.444301 + 0.895877i \(0.646548\pi\)
\(198\) 0 0
\(199\) 18.4721 1.30945 0.654727 0.755865i \(-0.272783\pi\)
0.654727 + 0.755865i \(0.272783\pi\)
\(200\) 0 0
\(201\) 1.52786 0.107767
\(202\) 0 0
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) 9.70820 0.678050
\(206\) 0 0
\(207\) 2.47214 0.171825
\(208\) 0 0
\(209\) −1.23607 −0.0855006
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.23607 −0.357097
\(216\) 0 0
\(217\) −46.8328 −3.17922
\(218\) 0 0
\(219\) 12.4721 0.842789
\(220\) 0 0
\(221\) 3.41641 0.229812
\(222\) 0 0
\(223\) 22.4721 1.50485 0.752423 0.658680i \(-0.228885\pi\)
0.752423 + 0.658680i \(0.228885\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −21.4164 −1.41524 −0.707618 0.706595i \(-0.750230\pi\)
−0.707618 + 0.706595i \(0.750230\pi\)
\(230\) 0 0
\(231\) −6.47214 −0.425835
\(232\) 0 0
\(233\) 23.8885 1.56499 0.782495 0.622657i \(-0.213947\pi\)
0.782495 + 0.622657i \(0.213947\pi\)
\(234\) 0 0
\(235\) −2.47214 −0.161264
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −11.7082 −0.757341 −0.378670 0.925532i \(-0.623619\pi\)
−0.378670 + 0.925532i \(0.623619\pi\)
\(240\) 0 0
\(241\) 6.94427 0.447320 0.223660 0.974667i \(-0.428199\pi\)
0.223660 + 0.974667i \(0.428199\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −20.4164 −1.30436
\(246\) 0 0
\(247\) 0.763932 0.0486078
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 3.70820 0.234060 0.117030 0.993128i \(-0.462663\pi\)
0.117030 + 0.993128i \(0.462663\pi\)
\(252\) 0 0
\(253\) −3.05573 −0.192112
\(254\) 0 0
\(255\) −4.47214 −0.280056
\(256\) 0 0
\(257\) −22.9443 −1.43122 −0.715612 0.698498i \(-0.753852\pi\)
−0.715612 + 0.698498i \(0.753852\pi\)
\(258\) 0 0
\(259\) 16.9443 1.05287
\(260\) 0 0
\(261\) 0.763932 0.0472862
\(262\) 0 0
\(263\) 3.05573 0.188424 0.0942121 0.995552i \(-0.469967\pi\)
0.0942121 + 0.995552i \(0.469967\pi\)
\(264\) 0 0
\(265\) 0.472136 0.0290031
\(266\) 0 0
\(267\) −1.70820 −0.104540
\(268\) 0 0
\(269\) 5.70820 0.348035 0.174018 0.984743i \(-0.444325\pi\)
0.174018 + 0.984743i \(0.444325\pi\)
\(270\) 0 0
\(271\) −2.47214 −0.150172 −0.0750858 0.997177i \(-0.523923\pi\)
−0.0750858 + 0.997177i \(0.523923\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) −1.23607 −0.0745377
\(276\) 0 0
\(277\) −16.4721 −0.989715 −0.494857 0.868974i \(-0.664780\pi\)
−0.494857 + 0.868974i \(0.664780\pi\)
\(278\) 0 0
\(279\) −8.94427 −0.535480
\(280\) 0 0
\(281\) −15.2361 −0.908908 −0.454454 0.890770i \(-0.650166\pi\)
−0.454454 + 0.890770i \(0.650166\pi\)
\(282\) 0 0
\(283\) −28.0689 −1.66852 −0.834261 0.551370i \(-0.814105\pi\)
−0.834261 + 0.551370i \(0.814105\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) −50.8328 −3.00057
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) −4.76393 −0.279267
\(292\) 0 0
\(293\) 3.88854 0.227171 0.113586 0.993528i \(-0.463766\pi\)
0.113586 + 0.993528i \(0.463766\pi\)
\(294\) 0 0
\(295\) 10.4721 0.609711
\(296\) 0 0
\(297\) −1.23607 −0.0717239
\(298\) 0 0
\(299\) 1.88854 0.109217
\(300\) 0 0
\(301\) 27.4164 1.58026
\(302\) 0 0
\(303\) −13.4164 −0.770752
\(304\) 0 0
\(305\) −4.47214 −0.256074
\(306\) 0 0
\(307\) −17.5279 −1.00037 −0.500184 0.865919i \(-0.666734\pi\)
−0.500184 + 0.865919i \(0.666734\pi\)
\(308\) 0 0
\(309\) 9.52786 0.542021
\(310\) 0 0
\(311\) 19.7082 1.11755 0.558775 0.829319i \(-0.311272\pi\)
0.558775 + 0.829319i \(0.311272\pi\)
\(312\) 0 0
\(313\) −23.8885 −1.35026 −0.675130 0.737699i \(-0.735913\pi\)
−0.675130 + 0.737699i \(0.735913\pi\)
\(314\) 0 0
\(315\) −5.23607 −0.295019
\(316\) 0 0
\(317\) 12.4721 0.700505 0.350252 0.936655i \(-0.386096\pi\)
0.350252 + 0.936655i \(0.386096\pi\)
\(318\) 0 0
\(319\) −0.944272 −0.0528691
\(320\) 0 0
\(321\) −8.94427 −0.499221
\(322\) 0 0
\(323\) 4.47214 0.248836
\(324\) 0 0
\(325\) 0.763932 0.0423753
\(326\) 0 0
\(327\) −4.47214 −0.247310
\(328\) 0 0
\(329\) 12.9443 0.713641
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 3.23607 0.177335
\(334\) 0 0
\(335\) −1.52786 −0.0834761
\(336\) 0 0
\(337\) −28.7639 −1.56687 −0.783436 0.621473i \(-0.786535\pi\)
−0.783436 + 0.621473i \(0.786535\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 11.0557 0.598701
\(342\) 0 0
\(343\) 70.2492 3.79310
\(344\) 0 0
\(345\) −2.47214 −0.133095
\(346\) 0 0
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) 10.9443 0.585833 0.292917 0.956138i \(-0.405374\pi\)
0.292917 + 0.956138i \(0.405374\pi\)
\(350\) 0 0
\(351\) 0.763932 0.0407757
\(352\) 0 0
\(353\) −14.9443 −0.795403 −0.397702 0.917515i \(-0.630192\pi\)
−0.397702 + 0.917515i \(0.630192\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 23.4164 1.23933
\(358\) 0 0
\(359\) −22.1803 −1.17063 −0.585317 0.810805i \(-0.699030\pi\)
−0.585317 + 0.810805i \(0.699030\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −9.47214 −0.497158
\(364\) 0 0
\(365\) −12.4721 −0.652821
\(366\) 0 0
\(367\) −29.2361 −1.52611 −0.763055 0.646333i \(-0.776302\pi\)
−0.763055 + 0.646333i \(0.776302\pi\)
\(368\) 0 0
\(369\) −9.70820 −0.505389
\(370\) 0 0
\(371\) −2.47214 −0.128347
\(372\) 0 0
\(373\) −7.23607 −0.374669 −0.187335 0.982296i \(-0.559985\pi\)
−0.187335 + 0.982296i \(0.559985\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 0.583592 0.0300565
\(378\) 0 0
\(379\) 1.88854 0.0970080 0.0485040 0.998823i \(-0.484555\pi\)
0.0485040 + 0.998823i \(0.484555\pi\)
\(380\) 0 0
\(381\) 20.0000 1.02463
\(382\) 0 0
\(383\) −9.88854 −0.505281 −0.252640 0.967560i \(-0.581299\pi\)
−0.252640 + 0.967560i \(0.581299\pi\)
\(384\) 0 0
\(385\) 6.47214 0.329851
\(386\) 0 0
\(387\) 5.23607 0.266164
\(388\) 0 0
\(389\) 32.8328 1.66469 0.832345 0.554258i \(-0.186998\pi\)
0.832345 + 0.554258i \(0.186998\pi\)
\(390\) 0 0
\(391\) 11.0557 0.559112
\(392\) 0 0
\(393\) 14.1803 0.715304
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −6.58359 −0.330421 −0.165211 0.986258i \(-0.552830\pi\)
−0.165211 + 0.986258i \(0.552830\pi\)
\(398\) 0 0
\(399\) 5.23607 0.262131
\(400\) 0 0
\(401\) −23.2361 −1.16035 −0.580177 0.814490i \(-0.697017\pi\)
−0.580177 + 0.814490i \(0.697017\pi\)
\(402\) 0 0
\(403\) −6.83282 −0.340367
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 4.47214 0.221133 0.110566 0.993869i \(-0.464734\pi\)
0.110566 + 0.993869i \(0.464734\pi\)
\(410\) 0 0
\(411\) 15.8885 0.783724
\(412\) 0 0
\(413\) −54.8328 −2.69815
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −14.4721 −0.708704
\(418\) 0 0
\(419\) −22.1803 −1.08358 −0.541790 0.840514i \(-0.682253\pi\)
−0.541790 + 0.840514i \(0.682253\pi\)
\(420\) 0 0
\(421\) −24.8328 −1.21028 −0.605139 0.796120i \(-0.706882\pi\)
−0.605139 + 0.796120i \(0.706882\pi\)
\(422\) 0 0
\(423\) 2.47214 0.120199
\(424\) 0 0
\(425\) 4.47214 0.216930
\(426\) 0 0
\(427\) 23.4164 1.13320
\(428\) 0 0
\(429\) −0.944272 −0.0455899
\(430\) 0 0
\(431\) 25.3050 1.21890 0.609448 0.792826i \(-0.291391\pi\)
0.609448 + 0.792826i \(0.291391\pi\)
\(432\) 0 0
\(433\) 0.763932 0.0367122 0.0183561 0.999832i \(-0.494157\pi\)
0.0183561 + 0.999832i \(0.494157\pi\)
\(434\) 0 0
\(435\) −0.763932 −0.0366277
\(436\) 0 0
\(437\) 2.47214 0.118258
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 20.4164 0.972210
\(442\) 0 0
\(443\) 19.4164 0.922501 0.461251 0.887270i \(-0.347401\pi\)
0.461251 + 0.887270i \(0.347401\pi\)
\(444\) 0 0
\(445\) 1.70820 0.0809766
\(446\) 0 0
\(447\) 17.4164 0.823768
\(448\) 0 0
\(449\) −9.70820 −0.458158 −0.229079 0.973408i \(-0.573571\pi\)
−0.229079 + 0.973408i \(0.573571\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 12.0000 0.563809
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 16.8328 0.787406 0.393703 0.919238i \(-0.371194\pi\)
0.393703 + 0.919238i \(0.371194\pi\)
\(458\) 0 0
\(459\) 4.47214 0.208741
\(460\) 0 0
\(461\) −8.47214 −0.394587 −0.197293 0.980344i \(-0.563215\pi\)
−0.197293 + 0.980344i \(0.563215\pi\)
\(462\) 0 0
\(463\) 25.5967 1.18958 0.594791 0.803880i \(-0.297235\pi\)
0.594791 + 0.803880i \(0.297235\pi\)
\(464\) 0 0
\(465\) 8.94427 0.414781
\(466\) 0 0
\(467\) −4.58359 −0.212103 −0.106052 0.994361i \(-0.533821\pi\)
−0.106052 + 0.994361i \(0.533821\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −21.4164 −0.986816
\(472\) 0 0
\(473\) −6.47214 −0.297589
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −0.472136 −0.0216176
\(478\) 0 0
\(479\) −14.1803 −0.647916 −0.323958 0.946071i \(-0.605014\pi\)
−0.323958 + 0.946071i \(0.605014\pi\)
\(480\) 0 0
\(481\) 2.47214 0.112720
\(482\) 0 0
\(483\) 12.9443 0.588985
\(484\) 0 0
\(485\) 4.76393 0.216319
\(486\) 0 0
\(487\) 4.58359 0.207702 0.103851 0.994593i \(-0.466883\pi\)
0.103851 + 0.994593i \(0.466883\pi\)
\(488\) 0 0
\(489\) 20.6525 0.933937
\(490\) 0 0
\(491\) −32.6525 −1.47359 −0.736793 0.676119i \(-0.763661\pi\)
−0.736793 + 0.676119i \(0.763661\pi\)
\(492\) 0 0
\(493\) 3.41641 0.153867
\(494\) 0 0
\(495\) 1.23607 0.0555571
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −11.4164 −0.511069 −0.255534 0.966800i \(-0.582251\pi\)
−0.255534 + 0.966800i \(0.582251\pi\)
\(500\) 0 0
\(501\) 13.8885 0.620494
\(502\) 0 0
\(503\) −12.9443 −0.577157 −0.288578 0.957456i \(-0.593183\pi\)
−0.288578 + 0.957456i \(0.593183\pi\)
\(504\) 0 0
\(505\) 13.4164 0.597022
\(506\) 0 0
\(507\) −12.4164 −0.551432
\(508\) 0 0
\(509\) 30.2918 1.34266 0.671330 0.741158i \(-0.265723\pi\)
0.671330 + 0.741158i \(0.265723\pi\)
\(510\) 0 0
\(511\) 65.3050 2.88892
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) −9.52786 −0.419848
\(516\) 0 0
\(517\) −3.05573 −0.134391
\(518\) 0 0
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) −26.2918 −1.15186 −0.575932 0.817497i \(-0.695361\pi\)
−0.575932 + 0.817497i \(0.695361\pi\)
\(522\) 0 0
\(523\) −0.944272 −0.0412901 −0.0206451 0.999787i \(-0.506572\pi\)
−0.0206451 + 0.999787i \(0.506572\pi\)
\(524\) 0 0
\(525\) 5.23607 0.228521
\(526\) 0 0
\(527\) −40.0000 −1.74243
\(528\) 0 0
\(529\) −16.8885 −0.734285
\(530\) 0 0
\(531\) −10.4721 −0.454452
\(532\) 0 0
\(533\) −7.41641 −0.321240
\(534\) 0 0
\(535\) 8.94427 0.386695
\(536\) 0 0
\(537\) −15.4164 −0.665267
\(538\) 0 0
\(539\) −25.2361 −1.08699
\(540\) 0 0
\(541\) −29.7771 −1.28022 −0.640108 0.768285i \(-0.721111\pi\)
−0.640108 + 0.768285i \(0.721111\pi\)
\(542\) 0 0
\(543\) 8.47214 0.363574
\(544\) 0 0
\(545\) 4.47214 0.191565
\(546\) 0 0
\(547\) −37.3050 −1.59504 −0.797522 0.603289i \(-0.793856\pi\)
−0.797522 + 0.603289i \(0.793856\pi\)
\(548\) 0 0
\(549\) 4.47214 0.190866
\(550\) 0 0
\(551\) 0.763932 0.0325446
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −3.23607 −0.137363
\(556\) 0 0
\(557\) 32.8328 1.39117 0.695586 0.718443i \(-0.255145\pi\)
0.695586 + 0.718443i \(0.255145\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −5.52786 −0.233387
\(562\) 0 0
\(563\) −9.88854 −0.416752 −0.208376 0.978049i \(-0.566818\pi\)
−0.208376 + 0.978049i \(0.566818\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 5.23607 0.219894
\(568\) 0 0
\(569\) −32.5410 −1.36419 −0.682095 0.731263i \(-0.738931\pi\)
−0.682095 + 0.731263i \(0.738931\pi\)
\(570\) 0 0
\(571\) −21.3050 −0.891584 −0.445792 0.895136i \(-0.647078\pi\)
−0.445792 + 0.895136i \(0.647078\pi\)
\(572\) 0 0
\(573\) −6.18034 −0.258187
\(574\) 0 0
\(575\) 2.47214 0.103095
\(576\) 0 0
\(577\) −9.05573 −0.376995 −0.188497 0.982074i \(-0.560362\pi\)
−0.188497 + 0.982074i \(0.560362\pi\)
\(578\) 0 0
\(579\) 5.70820 0.237225
\(580\) 0 0
\(581\) 20.9443 0.868915
\(582\) 0 0
\(583\) 0.583592 0.0241699
\(584\) 0 0
\(585\) −0.763932 −0.0315847
\(586\) 0 0
\(587\) −11.4164 −0.471206 −0.235603 0.971849i \(-0.575706\pi\)
−0.235603 + 0.971849i \(0.575706\pi\)
\(588\) 0 0
\(589\) −8.94427 −0.368542
\(590\) 0 0
\(591\) −12.4721 −0.513035
\(592\) 0 0
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 0 0
\(595\) −23.4164 −0.959979
\(596\) 0 0
\(597\) 18.4721 0.756014
\(598\) 0 0
\(599\) 47.4164 1.93738 0.968691 0.248270i \(-0.0798619\pi\)
0.968691 + 0.248270i \(0.0798619\pi\)
\(600\) 0 0
\(601\) −10.3607 −0.422621 −0.211310 0.977419i \(-0.567773\pi\)
−0.211310 + 0.977419i \(0.567773\pi\)
\(602\) 0 0
\(603\) 1.52786 0.0622194
\(604\) 0 0
\(605\) 9.47214 0.385097
\(606\) 0 0
\(607\) −33.5279 −1.36085 −0.680427 0.732816i \(-0.738206\pi\)
−0.680427 + 0.732816i \(0.738206\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 1.88854 0.0764023
\(612\) 0 0
\(613\) −35.5279 −1.43496 −0.717478 0.696581i \(-0.754704\pi\)
−0.717478 + 0.696581i \(0.754704\pi\)
\(614\) 0 0
\(615\) 9.70820 0.391473
\(616\) 0 0
\(617\) −39.3050 −1.58236 −0.791179 0.611585i \(-0.790532\pi\)
−0.791179 + 0.611585i \(0.790532\pi\)
\(618\) 0 0
\(619\) 32.3607 1.30069 0.650343 0.759641i \(-0.274625\pi\)
0.650343 + 0.759641i \(0.274625\pi\)
\(620\) 0 0
\(621\) 2.47214 0.0992034
\(622\) 0 0
\(623\) −8.94427 −0.358345
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.23607 −0.0493638
\(628\) 0 0
\(629\) 14.4721 0.577042
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) −20.0000 −0.793676
\(636\) 0 0
\(637\) 15.5967 0.617966
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −44.7639 −1.76807 −0.884035 0.467422i \(-0.845183\pi\)
−0.884035 + 0.467422i \(0.845183\pi\)
\(642\) 0 0
\(643\) 17.5967 0.693948 0.346974 0.937875i \(-0.387209\pi\)
0.346974 + 0.937875i \(0.387209\pi\)
\(644\) 0 0
\(645\) −5.23607 −0.206170
\(646\) 0 0
\(647\) 14.8328 0.583138 0.291569 0.956550i \(-0.405823\pi\)
0.291569 + 0.956550i \(0.405823\pi\)
\(648\) 0 0
\(649\) 12.9443 0.508107
\(650\) 0 0
\(651\) −46.8328 −1.83552
\(652\) 0 0
\(653\) −31.5279 −1.23378 −0.616890 0.787049i \(-0.711608\pi\)
−0.616890 + 0.787049i \(0.711608\pi\)
\(654\) 0 0
\(655\) −14.1803 −0.554072
\(656\) 0 0
\(657\) 12.4721 0.486584
\(658\) 0 0
\(659\) 30.8328 1.20108 0.600538 0.799596i \(-0.294953\pi\)
0.600538 + 0.799596i \(0.294953\pi\)
\(660\) 0 0
\(661\) 23.3050 0.906458 0.453229 0.891394i \(-0.350272\pi\)
0.453229 + 0.891394i \(0.350272\pi\)
\(662\) 0 0
\(663\) 3.41641 0.132682
\(664\) 0 0
\(665\) −5.23607 −0.203046
\(666\) 0 0
\(667\) 1.88854 0.0731247
\(668\) 0 0
\(669\) 22.4721 0.868823
\(670\) 0 0
\(671\) −5.52786 −0.213401
\(672\) 0 0
\(673\) −27.5967 −1.06378 −0.531888 0.846815i \(-0.678517\pi\)
−0.531888 + 0.846815i \(0.678517\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 15.5279 0.596784 0.298392 0.954443i \(-0.403550\pi\)
0.298392 + 0.954443i \(0.403550\pi\)
\(678\) 0 0
\(679\) −24.9443 −0.957273
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 31.7771 1.21592 0.607958 0.793969i \(-0.291989\pi\)
0.607958 + 0.793969i \(0.291989\pi\)
\(684\) 0 0
\(685\) −15.8885 −0.607070
\(686\) 0 0
\(687\) −21.4164 −0.817087
\(688\) 0 0
\(689\) −0.360680 −0.0137408
\(690\) 0 0
\(691\) −25.5279 −0.971126 −0.485563 0.874202i \(-0.661385\pi\)
−0.485563 + 0.874202i \(0.661385\pi\)
\(692\) 0 0
\(693\) −6.47214 −0.245856
\(694\) 0 0
\(695\) 14.4721 0.548959
\(696\) 0 0
\(697\) −43.4164 −1.64451
\(698\) 0 0
\(699\) 23.8885 0.903548
\(700\) 0 0
\(701\) −14.5836 −0.550815 −0.275407 0.961328i \(-0.588813\pi\)
−0.275407 + 0.961328i \(0.588813\pi\)
\(702\) 0 0
\(703\) 3.23607 0.122051
\(704\) 0 0
\(705\) −2.47214 −0.0931060
\(706\) 0 0
\(707\) −70.2492 −2.64199
\(708\) 0 0
\(709\) −31.3050 −1.17568 −0.587841 0.808976i \(-0.700022\pi\)
−0.587841 + 0.808976i \(0.700022\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −22.1115 −0.828081
\(714\) 0 0
\(715\) 0.944272 0.0353138
\(716\) 0 0
\(717\) −11.7082 −0.437251
\(718\) 0 0
\(719\) 1.23607 0.0460976 0.0230488 0.999734i \(-0.492663\pi\)
0.0230488 + 0.999734i \(0.492663\pi\)
\(720\) 0 0
\(721\) 49.8885 1.85795
\(722\) 0 0
\(723\) 6.94427 0.258260
\(724\) 0 0
\(725\) 0.763932 0.0283717
\(726\) 0 0
\(727\) 20.6525 0.765958 0.382979 0.923757i \(-0.374898\pi\)
0.382979 + 0.923757i \(0.374898\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 23.4164 0.866087
\(732\) 0 0
\(733\) −36.8328 −1.36045 −0.680226 0.733003i \(-0.738118\pi\)
−0.680226 + 0.733003i \(0.738118\pi\)
\(734\) 0 0
\(735\) −20.4164 −0.753071
\(736\) 0 0
\(737\) −1.88854 −0.0695654
\(738\) 0 0
\(739\) −8.94427 −0.329020 −0.164510 0.986375i \(-0.552604\pi\)
−0.164510 + 0.986375i \(0.552604\pi\)
\(740\) 0 0
\(741\) 0.763932 0.0280637
\(742\) 0 0
\(743\) −44.9443 −1.64885 −0.824423 0.565975i \(-0.808500\pi\)
−0.824423 + 0.565975i \(0.808500\pi\)
\(744\) 0 0
\(745\) −17.4164 −0.638088
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) −46.8328 −1.71123
\(750\) 0 0
\(751\) 8.94427 0.326381 0.163191 0.986595i \(-0.447821\pi\)
0.163191 + 0.986595i \(0.447821\pi\)
\(752\) 0 0
\(753\) 3.70820 0.135134
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) −2.94427 −0.107011 −0.0535057 0.998568i \(-0.517040\pi\)
−0.0535057 + 0.998568i \(0.517040\pi\)
\(758\) 0 0
\(759\) −3.05573 −0.110916
\(760\) 0 0
\(761\) 9.41641 0.341345 0.170672 0.985328i \(-0.445406\pi\)
0.170672 + 0.985328i \(0.445406\pi\)
\(762\) 0 0
\(763\) −23.4164 −0.847731
\(764\) 0 0
\(765\) −4.47214 −0.161690
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 8.11146 0.292507 0.146253 0.989247i \(-0.453279\pi\)
0.146253 + 0.989247i \(0.453279\pi\)
\(770\) 0 0
\(771\) −22.9443 −0.826318
\(772\) 0 0
\(773\) 39.5279 1.42172 0.710859 0.703334i \(-0.248306\pi\)
0.710859 + 0.703334i \(0.248306\pi\)
\(774\) 0 0
\(775\) −8.94427 −0.321288
\(776\) 0 0
\(777\) 16.9443 0.607872
\(778\) 0 0
\(779\) −9.70820 −0.347833
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0.763932 0.0273007
\(784\) 0 0
\(785\) 21.4164 0.764384
\(786\) 0 0
\(787\) 10.8328 0.386148 0.193074 0.981184i \(-0.438154\pi\)
0.193074 + 0.981184i \(0.438154\pi\)
\(788\) 0 0
\(789\) 3.05573 0.108787
\(790\) 0 0
\(791\) −10.4721 −0.372346
\(792\) 0 0
\(793\) 3.41641 0.121320
\(794\) 0 0
\(795\) 0.472136 0.0167449
\(796\) 0 0
\(797\) 37.7771 1.33813 0.669067 0.743202i \(-0.266694\pi\)
0.669067 + 0.743202i \(0.266694\pi\)
\(798\) 0 0
\(799\) 11.0557 0.391124
\(800\) 0 0
\(801\) −1.70820 −0.0603564
\(802\) 0 0
\(803\) −15.4164 −0.544033
\(804\) 0 0
\(805\) −12.9443 −0.456226
\(806\) 0 0
\(807\) 5.70820 0.200938
\(808\) 0 0
\(809\) 38.9443 1.36921 0.684604 0.728915i \(-0.259975\pi\)
0.684604 + 0.728915i \(0.259975\pi\)
\(810\) 0 0
\(811\) 0.944272 0.0331579 0.0165789 0.999863i \(-0.494723\pi\)
0.0165789 + 0.999863i \(0.494723\pi\)
\(812\) 0 0
\(813\) −2.47214 −0.0867016
\(814\) 0 0
\(815\) −20.6525 −0.723425
\(816\) 0 0
\(817\) 5.23607 0.183187
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 15.5279 0.541926 0.270963 0.962590i \(-0.412658\pi\)
0.270963 + 0.962590i \(0.412658\pi\)
\(822\) 0 0
\(823\) −2.18034 −0.0760019 −0.0380009 0.999278i \(-0.512099\pi\)
−0.0380009 + 0.999278i \(0.512099\pi\)
\(824\) 0 0
\(825\) −1.23607 −0.0430344
\(826\) 0 0
\(827\) 8.00000 0.278187 0.139094 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) 0 0
\(829\) 47.3050 1.64297 0.821484 0.570231i \(-0.193146\pi\)
0.821484 + 0.570231i \(0.193146\pi\)
\(830\) 0 0
\(831\) −16.4721 −0.571412
\(832\) 0 0
\(833\) 91.3050 3.16353
\(834\) 0 0
\(835\) −13.8885 −0.480633
\(836\) 0 0
\(837\) −8.94427 −0.309159
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −28.4164 −0.979876
\(842\) 0 0
\(843\) −15.2361 −0.524758
\(844\) 0 0
\(845\) 12.4164 0.427137
\(846\) 0 0
\(847\) −49.5967 −1.70416
\(848\) 0 0
\(849\) −28.0689 −0.963322
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 42.7214 1.46275 0.731376 0.681975i \(-0.238879\pi\)
0.731376 + 0.681975i \(0.238879\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 24.4721 0.835952 0.417976 0.908458i \(-0.362740\pi\)
0.417976 + 0.908458i \(0.362740\pi\)
\(858\) 0 0
\(859\) 10.8328 0.369611 0.184805 0.982775i \(-0.440834\pi\)
0.184805 + 0.982775i \(0.440834\pi\)
\(860\) 0 0
\(861\) −50.8328 −1.73238
\(862\) 0 0
\(863\) 46.8328 1.59421 0.797104 0.603842i \(-0.206364\pi\)
0.797104 + 0.603842i \(0.206364\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) 3.00000 0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 1.16718 0.0395485
\(872\) 0 0
\(873\) −4.76393 −0.161235
\(874\) 0 0
\(875\) −5.23607 −0.177011
\(876\) 0 0
\(877\) 37.7082 1.27332 0.636658 0.771146i \(-0.280316\pi\)
0.636658 + 0.771146i \(0.280316\pi\)
\(878\) 0 0
\(879\) 3.88854 0.131157
\(880\) 0 0
\(881\) −55.3050 −1.86327 −0.931636 0.363394i \(-0.881618\pi\)
−0.931636 + 0.363394i \(0.881618\pi\)
\(882\) 0 0
\(883\) 25.5967 0.861399 0.430700 0.902495i \(-0.358267\pi\)
0.430700 + 0.902495i \(0.358267\pi\)
\(884\) 0 0
\(885\) 10.4721 0.352017
\(886\) 0 0
\(887\) −25.8885 −0.869252 −0.434626 0.900611i \(-0.643119\pi\)
−0.434626 + 0.900611i \(0.643119\pi\)
\(888\) 0 0
\(889\) 104.721 3.51224
\(890\) 0 0
\(891\) −1.23607 −0.0414098
\(892\) 0 0
\(893\) 2.47214 0.0827269
\(894\) 0 0
\(895\) 15.4164 0.515314
\(896\) 0 0
\(897\) 1.88854 0.0630566
\(898\) 0 0
\(899\) −6.83282 −0.227887
\(900\) 0 0
\(901\) −2.11146 −0.0703428
\(902\) 0 0
\(903\) 27.4164 0.912361
\(904\) 0 0
\(905\) −8.47214 −0.281623
\(906\) 0 0
\(907\) 10.1115 0.335745 0.167873 0.985809i \(-0.446310\pi\)
0.167873 + 0.985809i \(0.446310\pi\)
\(908\) 0 0
\(909\) −13.4164 −0.444994
\(910\) 0 0
\(911\) −43.7771 −1.45040 −0.725200 0.688538i \(-0.758253\pi\)
−0.725200 + 0.688538i \(0.758253\pi\)
\(912\) 0 0
\(913\) −4.94427 −0.163632
\(914\) 0 0
\(915\) −4.47214 −0.147844
\(916\) 0 0
\(917\) 74.2492 2.45193
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) −17.5279 −0.577563
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.23607 0.106401
\(926\) 0 0
\(927\) 9.52786 0.312936
\(928\) 0 0
\(929\) 27.8885 0.914993 0.457497 0.889211i \(-0.348746\pi\)
0.457497 + 0.889211i \(0.348746\pi\)
\(930\) 0 0
\(931\) 20.4164 0.669121
\(932\) 0 0
\(933\) 19.7082 0.645218
\(934\) 0 0
\(935\) 5.52786 0.180780
\(936\) 0 0
\(937\) −37.4164 −1.22234 −0.611170 0.791499i \(-0.709301\pi\)
−0.611170 + 0.791499i \(0.709301\pi\)
\(938\) 0 0
\(939\) −23.8885 −0.779573
\(940\) 0 0
\(941\) −17.1246 −0.558246 −0.279123 0.960255i \(-0.590044\pi\)
−0.279123 + 0.960255i \(0.590044\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) −5.23607 −0.170329
\(946\) 0 0
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) 9.52786 0.309288
\(950\) 0 0
\(951\) 12.4721 0.404437
\(952\) 0 0
\(953\) 42.9443 1.39110 0.695551 0.718477i \(-0.255160\pi\)
0.695551 + 0.718477i \(0.255160\pi\)
\(954\) 0 0
\(955\) 6.18034 0.199991
\(956\) 0 0
\(957\) −0.944272 −0.0305240
\(958\) 0 0
\(959\) 83.1935 2.68646
\(960\) 0 0
\(961\) 49.0000 1.58065
\(962\) 0 0
\(963\) −8.94427 −0.288225
\(964\) 0 0
\(965\) −5.70820 −0.183754
\(966\) 0 0
\(967\) −18.1803 −0.584640 −0.292320 0.956321i \(-0.594427\pi\)
−0.292320 + 0.956321i \(0.594427\pi\)
\(968\) 0 0
\(969\) 4.47214 0.143666
\(970\) 0 0
\(971\) −38.8328 −1.24620 −0.623102 0.782140i \(-0.714128\pi\)
−0.623102 + 0.782140i \(0.714128\pi\)
\(972\) 0 0
\(973\) −75.7771 −2.42930
\(974\) 0 0
\(975\) 0.763932 0.0244654
\(976\) 0 0
\(977\) 30.5836 0.978456 0.489228 0.872156i \(-0.337279\pi\)
0.489228 + 0.872156i \(0.337279\pi\)
\(978\) 0 0
\(979\) 2.11146 0.0674824
\(980\) 0 0
\(981\) −4.47214 −0.142784
\(982\) 0 0
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) 0 0
\(985\) 12.4721 0.397395
\(986\) 0 0
\(987\) 12.9443 0.412021
\(988\) 0 0
\(989\) 12.9443 0.411604
\(990\) 0 0
\(991\) 28.9443 0.919445 0.459723 0.888063i \(-0.347949\pi\)
0.459723 + 0.888063i \(0.347949\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −18.4721 −0.585606
\(996\) 0 0
\(997\) 1.41641 0.0448581 0.0224290 0.999748i \(-0.492860\pi\)
0.0224290 + 0.999748i \(0.492860\pi\)
\(998\) 0 0
\(999\) 3.23607 0.102385
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2280.2.a.p.1.2 2
3.2 odd 2 6840.2.a.bd.1.2 2
4.3 odd 2 4560.2.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.p.1.2 2 1.1 even 1 trivial
4560.2.a.be.1.1 2 4.3 odd 2
6840.2.a.bd.1.2 2 3.2 odd 2